Nonuniform Sampling Theory and Practice

Information Technology: Transmission, Processing, and Storage

Series Editor: Jack Keil Wolf University of California at San Diego La Jolla, California

Editorial Board: James E. Mazo Bell Laboratories, Lucent Technologies Murray Hill, New Jersey

John Proakis Northeastern University Boston, Massachusetts

William H. Tranter Virginia Polytechnic Institute and State University Blacksburg, Virginia

Multi-Carrier Digital Communications: Theory and Applications of OFDM Ahmad R. S. Bahai and Burton R. Saltzberg

Nonuniform Sampling: Theory and Practice Edited by Farokh Marvasti

Principles of Digital Transmission: With Wireless Applications Sergio Benedetto and Ezio Biglieri

Simulation of Communication Systems, Second Edition: Methodology, Modeling, and Techniques Michel C. Jeruchim, Philip Balaban, and K. Sam Shanmugan

A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher.

Nonuniform Sampling Theory and Practice

Edited by

Farokh Marvasti King's College

London, United Kingdom and

Sharif College of Technology Tehran, [ran

Springer Science+Business Media, LLC

Library of Congress Cataloging-in-Publication Data

Marvasti, Farokh A. Nonuniform sampling: theory and practice/Farokh A. Marvasti.

p. cm. -(Information technology: transmission, processing, and storage) Includes bibliographical references and index. ISBN 0-306-46445-4

1. Signal theory (Telecommunication)-Mathematics. 2. Sampling. 3. Time-series analysis. 1. Title. II. Series.

TK5102.5 .. M2953 2000 621.382'23-dc21

Additional material to book can be downloaded from http://extra.springer.com.

00-062213

ISBN 978-1-4613-5451-2 ISBN 978-1-4615-1229-5 (eBook)

DOI 10.1007/978-1-4615-1229-5

©2001 Springer Science+Business Media New York Originally published by Kluwer Academic/Plenum Publishers, New York in 2001

http://www.wkap.nll

ro 9 8 7 6 5 4 3 2 1

A C.I.P. record for this book is available from the Library of Congress

All rights reserved

No pari of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher.

Dedicated to:

The late Prof. 1. L. Yen who introduced interpolation formulae from special nonuniform samples l

The late Claude Shannon, father of information theory, who popularized the field of sampling theory in the engineering communi!)?

The Portuguese government, University of Aveiro, and the Portuguese people who hosted me while I was writing several chapters of this book

My family, Maryam, Salman, Laleh, Ali, and Narges for their understanding and help

My mother who instilled motivation and curiosity in me

My late father who cared so much about education

1 See his biography and the photo in Chapter 1. 2See more comments and photos in Chapters 1 and 2.

Contributors

Andreas Austeng • University of Oslo, Oslo, Norway

Sonali Bagchi • Mobilian Corporation, Hillsboro, OR, USA

John Thomas Barnett • Space and Naval Warfare Systems Center, San Diego, CA, USA

John Benedetto • University of Maryland, College Park, MD, USA

M. Bourgeois • Universite LYON I-CPE, Villeurbanne, France

P. L. Butzer • RWTH Aachen, Aachen, Germany

Willem L. de Koning •

A. 1. W. Duijndam • Paulo 1. S. G. Ferreira • D. Graveron-Demilly • Karlheinz Grochenig • Mohammed Hasan • C. O. H. Hindriks •

Delft University of Technology, Delft, The Netherlands

Philips Medical Systems, The Netherlands

Universadade de Aveiro, Aveiro, Portugal

Universite LYON I-CPE, Villeurbanne, France

University of Connecticut, Storrs, CT, USA

King's College London, London, UK

Delft University of Technology, Delft, The Netherlands

Sverre Holm • University of Oslo, Oslo, Norway

Jon-Fredrik Hopperstad • University of Oslo, Oslo, Norway

Karnran Iranpour • University of Oslo, Oslo, Norway

Timo I. Laakso • Helsinki University of Technology, Espoo, Finland

B. Lacaze • Farokh Marvasti

Lab TeSA, Cedex, France

• King's College London, London, UK & Sharif University of Technology, Tehran, Iran

vii

viii Contributors

Sanjit K. Mitra • University of California at Santa Barbara, Santa Barbara, CA, USA

Dale H. Mugler • University of Akron, Akron, OH, USA

M. Sandler • King's College London, London, UK

G. Schmeisser • University of Erlangen-Nurenberg, Erlangen, Germany

M. A. Schonewille • PGS Onshore, Houston, TX, USA

Sherry Scott • University of Maryland, College Park, MD, USA

Atif Sharaf • King's College London, London, UK

R. L. Stens • RWTH Aachen, Aachen, Germany

G. H. L. A. Stijnrnan • Delft University of Technology, Delft, The Netherlands

Thomas Strohmer • University of California at Davis, Davis, CA, USA

Vesa Vlilimiiki

D. van Ormont

• •

Helsinki University of Technology, Espoo, Finland

Delft University of Technology, Delft, The Netherlands

L. Gerard van Willigenburg • Wageningen University, Wageningen, The Netherlands

F. T. A. W Wajer

YanWu • Ahmed I. Zayed

• Delft University of Technology, Delft, The Netherlands

University of Akron, Akron, OH, USA

• University of Central Florida, Orlando, FL, USA

Foreword

I was very pleased when Farokh Marvasti asked me to write a Foreword to this volume. I can certainly recommend it to the scientific community in the confidence that its readers will get a real sense of the flavour, style and innovative character of nonuniform sampling and its applications.

The importance of sampling as a scientific principle, both in theory and practice, can hardly be in doubt. The reader who cares to glance through the Table of Contents cannot fail to be convinced as to the ubiquity of the subject and its broad scope; and if we allow that the subject finds its root in the finite interpolation problems that were already being studied in the middle part of the seventeenth century (and there are cogent reasons for taking this view), nonuniformity of the distribution of sample points has been present in sampling from its very beginnings.

A basic idea, one that lies at the foundations of sampling, is that it is very convenient to consider a signal, or function, as consisting merely of a collection of discrete samples, that is, values taken by the function at some countable set of sample points. When this can be done one is saying effectively that the information contained in the function's samples is equivalent to, or at least approximately equivalent to, that present in the whole function. In order to reach such a desirable position one needs to ask whether the set of sample points is a set of uniqueness for some class of functions, and if so, how a member of the class could be reconstructed from data which would often consist of samples of the function, or perhaps of pre-processed versions of it. Going beyond the finite case to that of infinitely many sample points, the simplest, indeed the "classical", case involves uniformly distributed sample points along the real line and is embodied in the famous sampling theorem associated with the names of E. T. Whittaker, V. A. Kotel'nikov and C. E. Shannon (and others); a theorem that can be found in many places throughout this book.

It is from such basic ideas that sampling developed, ever more rapidly, throughout the last century and into the present. Indeed, sampling is now a multi­disciplinary activity, and is often found where some or all of: harmonic analysis including classical Fourier analysis, function theory, approximation theory, prediction theory, stochastic processes, information theory and practical consid­erations of signal processing are seen to overlap (and that is just a short list!).

ix

x Foreword

Movements of scientific thought develop continuously, and are indeed intelligible to us as continuity. The past informs the present, which in turn will inevitably inform the future. However, we live in an era when these developments are, more often than not, accompanied by changes that are huge and rapid. We are forced, therefore, to ask the question: how do we cope with this change; how are we to deal with the new? I believe that books such as the present one have an important role to play here (and it will surely be a sad day for the human race when there is no role for books to play any more). We can pause with them; we can gain breathing space, a time to ponder, a time to sift foreground from background before the tidal waves of change press us forward once again.

On a more day-to-day level, the usefulness of a book such as this will depend in part on its selection from the masses of material available in· the literature and in the minds of its contributors; in part on how it improves access to the subject, and in part on whether its contributions contain careful analysis, informative bibliographies and a high quality of elucidation. I confidently predict that this book will not be found wanting in any of these attributes.

It seems that there is no shortage of new and interesting problems in non­uniform sampling, and I am sure that the present volume will help to maintain that interest.

Happy sampling! J. R. Higgins, Cambridge, UK

Preface

Shannon never claimed any credit for originating The Sampling Theorem, he simply realized as part of his original development of a mathematical Theory of Communication that a band-limited signal could (in theory) be reconstructed from uniformly spaced values provided the samples were closer than a critical amount. In my first encounters with The Sampling Theorem, it was called the "Shannon-Whittaker-Kotel'nikov Sampling Theorem", with honors shared by Shannon with the radar pioneer and originator of Signal Detection Theory. Regardless of the origins of the result-which are considered in this book-these three, and Shannon in particular, had a profound effect on the rapid spread of both understanding and applications of sampling to communications and signal processing theory and practice.

On the theoretical side, sampling provided the means of converting contin­uous time signals to discrete-time signals without loss of information. This permitted tools for discrete time signals such as linear algebra and time-series analysis to be applied to the evaluation of channel capacities and source rates of continuous time processes. When sampling, the discretization of time, was combined with quantization, the discretization of amplitude, the result was analog-to-digital conversion and the true beginnings of the modem digital revolution, as embodied in the first digital communication technique for contin­uous waveforms-pulse coded modulation (PCM), as popularized by Oliver, Pierce, and Shannon. The descendants of this technique are ubiquitous in the Internet and modem wireless communication.

Sampling has grown in many directions, especially nonuniform sampling and generalization incorporating transforms such as Fourier, Karhunen-Loeve, wavelets, and filter-banks. For all of these extensions, however, the basic issue for an engineer remains the discretization of time or space, whether the goal be to prove a theorem or transmit speech or video. Along with quantization, sampling resides at the border between the analog world of nature, and the digital world of communication, signal processing and computing. This book provides a thorough and varied tour of that border.

Prof Robert M. Gray Department of Electrical and Computer Engineering

Stanford University Stanford, California

xi

Contents

Chapter 1. Introduction F. Marvasti

1.1. Preliminaries...................................... 1.2. Historical Perspective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3. Summary of the Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4. Acknowledgment................................... 13

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Chapter 2. An Introduction to Sampling Analysis P. L. Butzer, G. Schmeisser, and R. L. Stens

2.1. Introduction and Some History. . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.1. Prelude.................................... 17 2.1.2. Some Further History. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1.3. Some Literature, Aims and Contents. . . . . . . . . . . . . . . . . . . 28

2.2. Foundations in Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.1. Fourier Transfonns, Fourier Series . . . . . . . . . . . . . . . . . . . . 32 2.2.2. Elementary Functions and Distributions. . . . . . . . . . . . . . . . . 41

2.3. The Sampling Theorem Itself. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3.1. Two Simple Proofs and an Engineering Demonstration. . . . . . . . 49 2.3.2. Four Further Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.4. Sampling Representations for Derivatives and Hilbert Transfonns . . . . . . 59 2.5. Multi-Channel Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.6. Sampling Theory for Non-Band-Limited Functions. . . . . . . . . . . . . . . 63

2.6.1. The General Situation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.6.2. Duration-Limited Signals. . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.7. Error Analysis in Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.7.1. Errors for Band-Limited Signals. . . . . . . . . . . . . . . . . . . . . 67 2.7.2. Errors for Not-Necessarily Band-Limited Signals. . . . . . . . . . . 71

2.8. Generalized Sampling Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

xiii

xiv Contents

2.8.1. General Convergence Theorems . . . . . . . . . . . . . . . . . . . .. 73 2.8.2. Rate of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76 2.8.3. Applications................................. 79

2.9. Connections of the Sampling Theorem with other Basic Theorems. . . .. 84 2.9.1. Band-Limited Situation. . . . . . . . . . . . . . . . . . . . . . . . . .. 85 2.9.2. General Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86

2.10. Sampling and Partial Fraction Decomposition. . . . . . . . . . . . . . . . .. 88 2.10.1. Cauchy's Partial Fraction Decomposition. . . . . . . . . . . . . .. 88 2.10.2. An Improvement of Cauchy's Method. . . . . . . . . . . . . . . .. 94 2.10.3. The Mittag-Leffler Theorem. . . . . . . . . . . . . . . . . . . . . .. 96

2.11. Sampling and Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98 2.11.1. Quadrature of Band-Limited Signals over the Real Line . . . . .. 98 2.11.2. Generalized Gaussian Quadrature Formulae. . . . . . . . . . . . .. 99 2.11.3. Quadrature of Non-Band-Limited Signals over the Real Line. .. 102 2.11.4. Weighted Quadrature over the Real Line . . . . . . . . . . . . . .. 106 2.11.5. Quadrature over a Semi-Infinite Interval. . . . . . . . . . . . . . .. 108 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 110 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 111

Chapter 3. Lagrange Interpolation and Sampling Theorems A. I. Zayed and P. 1. Butzer

3.1. Introduction............................... . ...... 123 3.2. Lagrange (Polynomial) Interpolation. . . . . . . . . . . . . . . . . . . . . . .. 124 3.3. Lagrange-Type Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 128

3.3.1. Uniform Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 128 3.3.2. Nonuniform Sampling . . . . . . . . . . . . . . . . . . . . . . . . . .. 137

3.4. Sampling Expansions Involving Derivatives. . . . . . . . . . . . . . . . . .. 140 3.5. Kramer's Sampling Theorem and Lagrange Interpolation. . . . . . . . . .. 146

3.5.1. Kramer's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 146 3.5.2. Connections with Boundary-Value Problems. . . . . . . . . . . . .. 148 3.5.3. Connections with Lagrange-Type Interpolation. . . . . . . . . . . .. 150 3.5.4. Generalization of Kramer's Theorem. . . . . . . . . . . . . . . . . .. 154

3.6. Boundary-Value Problems and Lagrange-Type Interpolation. . . . . . . . .. 158 3.6.1. The Main Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 158 3.6.2. Examples of Boundary-Value Problems Having the Lagrange

Interpolation Property. . . . . . . . . . . . . . . . . . . . . . . . . . .. 160 3.7. Extension of Kramer's Theorem to Approximately Band-Limited Signals.. 164

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 167

Chapter 4. Random Topics in Nonuniform Sampling F. Marvasti

Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 169 4.1. General Nonuniform Sampling Theorems. . . . . . . . . . . . . . . . . . . .. 170

Contents xv

4.1.1. Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . .. 171 4.2. Lagrange Interpolation ............................. " 173

4.2.1. Lagrange Interpolation for 2-D Signals. . . . . . . . . . . . . . .. 174 4.2.2. Lagrange Interpolation in Polar Coordinates ........... " 176

4.3. Nonunifonn Sampling for Non-Band-Limited Signals . . . . . . . . . . .. 179 4.3.1. Jittered Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 179

4.4. Past Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 181 4.4.1. Stability of Nonunifonn Sampling Interpolation . . . . . . . . . .. 182

4.5. Spectral Analysis of Nonunifonn Samples . . . . . . . . . . . . . . . . . .. 183 4.5.1. Extension of Parse val Relationship to Nonunifonn Samples. . .. 183 4.5.2. Spectral Analysis of Nonunifonn Samples for 2-D Signals ... " 185 4.5.3. The Spectrum of Nonunifonn Samples for I-D Signals. . . . . .. 189

4.6. Practical Reconstruction Techniques. . . . . . . . . . . . . . . . . . . . . .. 191 4.6.1. A Time Varying (also called Nonlinear) Method .... : . . . . .. 191 4.6.2. An Iterative Recovery Method. . . . . . . . . . . . . . . . . . . . .. 193

4.7. The Generalized Sampling Theorem, Filter Banks, QMF, Wavelets and Sub-Band Coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 199 4.7.1. Introduction................................. 199 4.7.2. Interpolation of Various Sampling Schemes. . . . . . . . . . . . .. 201 4.7.3. Periodic Nonunifonn Sampling. . . . . . . . . . . . . . . . . . . .. 203 4.7.4. Conclusion................................. 204

4.8. Interpolation of Lowpass Signals at Half the Nyquist Rate. . . . . . . . .. 204 4.8.1. Introduction................................. 205 4.8.2. Choice of Sampling Set. . . . . . . . . . . . . . . . . . . . . . . . .. 205 4.8.3. Lagrange Interpolation . . . . . . . . . . . . . . . . . . . . . . . . .. 206 4.8.4. Conclusion................................. 208

4.9. Nonunifonn Sampling Theorems for Bandpass Signals. . . . . . . . . . .. 208 4.9.1. Introduction................................. 208 4.9.2. Signal Recovery from the Nonuniform Samples of a Bandpass

Signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 208 4.9.3. Practical Methods to Find the Sampling Set Close to Half the

Nyquist Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 210 4.9.4. Conclusion...... . .......................... 211

4.10. Extremum Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 211 4.10.1. Introduction................................ 211 4.10.2. Description of Extremum Sampling. . . . . . . . . . . . . . . . . . 211 4.10.3. Discrete Finite Dimensional eN Signals. . . . . . . . . . . . . .. 212 4.1 0.4. Lagrange Interpolation (LI). . . . . . . . . . . . . . . . . . . . . .. 213 4.10.5. Matrix Approximation for the Extremum Reconstruction

(MAER) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 215 4.10.6. Modified Lagrange with Derivative Sampling (MLDS) . . . . .. 217 4.1 0.7. Discrete Lagrange Interpolation (DLI) . . . . . . . . . . . . . . .. 217 4.10.8. Iterative Methods (1M). . . . . . . . . . . . . . . . . . . . . . . . .. 220 4.10.9. Spline Interpolation (Ispline). . . . . . . . . . . . . . . . . . . . .. 221 4.10.10. Approximations with Other Interpolating Kernels and

Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 221

xvi Contents

4.l0.l1. Bipolar Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 224 4.l0.l2. Extremum and Zero-Crossing Positions . . . . . . . . . . . . . .. 225 4.l0.13. Reed Solomon (RS) Decoding. . . . . . . . . . . . . . . . . . . .. 227 4.l0.14. Time Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 229 4.l0.15. L2 Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 229 4.10.16. Comparison of Extremum Sampling to the Uniform Sampling.. 230

4.11. Random Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 231 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 232

Chapter 5. Iterative and Noniterative Recovery of Missing Samples for I-D Band-Limited Signals p. 1. S. G. Ferreira

5.1. Introduction...................................... 235 5.1.1. Outline of the Chapter. . . . . . . . . . . . . . . . . . . . . . . . . .. 236

5.2. Band-Limited Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 238 5.3. Iterative Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 242

5.3.1. The Papoulis-Gerchberg Iteration . . . . . . . . . . . . . . . . . . .. 242 5.3.2. Alternating Projections, POCS, and the Frame Algorithm. . . . .. 244 5.3.3. Constrained Iterative Restoration, Relaxation. . . . . . . . . . . . .. 244 5.3.4. Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .. 246 5.3.5. Upper and Lower Error Bounds. . . . . . . . . . . . . . . . . . . . .. 249 5.3.6. Other Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 254

5.4. Minimum Dimension Formulations . . . . . . . . . . . . . . . . . . . . . . .. 256 5.4.1. Examples and Comparisons. . . . . . . . . . . . . . . . . . . . . . .. 261

5.5. Stability Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 268 5.5.1. Examples................................... 276

5.6. Conclusions...................................... 277 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 278

Chapter 6. Numerical and Theoretical Aspects of Nonuniform Sampling of Band-Limited Images K. Grochenig and T. Strohmer

Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 283 6.1. Introduction...................................... 283 6.2. Band-Limited Images and Finite-Dimensional Models . . . . . . . . . . . .. 284

6.2.1. Band-Limited Images. . . . . . . . . . . . . . . . . . . . . . . . . . .. 285 6.2.2. Discrete Band-Limited Images . . . . . . . . . . . . . . . . . . . . .. 285 6.2.3. Trigonometric Polynomials. . . . . . . . . . . . . . . . . . . . . . . .. 286

6.3. Numerical Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 289 6.3.1. Vandermonde Systems. . . . . . . . . . . . . . . . . . . . . . . . . .. 289 6.3.2. Frames in Nonuniform Sampling. . . . . . . . . . . . . . . . . . . .. 290

Contents xvii

6.3.3. Toeplitz Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 291 6.3.4. An Efficient Reconstruction Algorithm. . . . . . . . . . . . . . . .. 293 6.3.5. Adaptive Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 297

6.4. Theoretical Back-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 299 6.4.1. Condition Numbers for Toeplitz Systems. . . . . . . . . . . . . . .. 299 6.4.2. Towards Nonuniform Sampling of Band-Limited Images . . . . .. 301 6.4.3. Sampling Theorems for Band-Limited Images. . . . . . . . . . . .. 304

6.5. Deeper Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 305 6.5.1. Bandwidth Estimation - A Multi-Level Approach. . . . . . . . . .. 305 6.5.2. Ill-Conditioned Sampling Problems . . . . . . . . . . . . . . . . . .. 309 6.5.3. Additional Knowledge About the Signal. . . . . . . . . . . . . . .. 311

6.6. Applications...................................... 312 6.6.1. . Object Boundary Recovery in Echocardiography . . . . . . . . . .. 313 6.6.2. Image Reconstruction in Exploration Geophysics . . . . . . . . . .. 315 6.6.3. Reconstruction of Missing Pixels in Images . . . . . . . . . . . . .. 318 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 322

Chapter 7. The Nonuniform Discrete Fourier Transform S. Bagchi and S. K. Mitra

Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 7.1. Introduction............................. . ... . ... . 326 7.2. The I-D NDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 326

7.2.1. Definition................................... 326 7.2.2. The Inverse NDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 327 7.2.3. Computation of the NDFT. . . . . . . . . . . . . . . . . . . . . . . .. 329

7.3. The 2-D NDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 334 7.3.1. Definition................................... 334 7.3.2. Special Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 335

7.4. I-D FIR Filter Design Using the NDFT . . . . . . . . . . . . . . . . . . . .. 340 7.4.1. Nonuniform Frequency Sampling Design Method. . . . . . . . . .. 340 7.4.2. Lowpass Filter Design. . . . . . . . . . . . . . . . . . . . . . . . . .. 341 7.4.3 A Design Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 345 7.4.4. Remarks.................................... 348

7.5. 2-D FIR Filter Design Using the NDFT . . . . . . . . . . . . . . . . . . . .. 349 7.5.1. Design Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 349 7.5.2. Nonuniform Frequency Sampling Design Method. . . . . . . . . .. 350 7.5.3. Diamond Filter Design. . . . . . . . . . . . . . . . . . . . . . . . . .. 351 7.5.4. A Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 355

7.6. Summary........................................ 358 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 358

,xviii Contents

Chapter 8. Reconstruction of Stationary Processes Sampled at Random Times B. Lacaze

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 361 8.1. Introduction................. . ..................... 361

8.1.1. Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . .. , 361 8.1.2. Estimator Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 362 8.1.3. Error...................................... 364 8.1.4. Error Minimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 366

8.2. Previous Work on Random Sampling. . . . . . . . . . . . . . . . . . . . . . .. 367 8.2.1. Interpolation Formula for Random Processes . . . . . . . . . . . . .. 367 8.2.2. Power Spectra Measurements. . . . . . . . . . . . . . . . . . . . . . .. 368

8.3. The Observed Random Case. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 369 8.3.1. The Finite Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 369 8.3.2. An Example in the Infinite Observed Case. . . . . . . . . . . . . . .. 371 8.3.3. The Band-Limited Infinite Case . . . . . . . . . . . . . . . . . . . . .. 372

8.4. The Unobserved Random Case. . . . . . . . . . . . . . . . . . . . . . . . . . .. 374 8.4.1. The Jitter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 374 8.4.2. The Best Estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 375 8.4.3. Two Particular Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 377 8.4.4. Examples.................................... 377

8.5. The Complex Process Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 381 8.5.1. A Remark about Uniform Sampling. . . . . . . . . . . . . . . . . . .. 381 8.5.2. Example of Random Sampling. . . . . . . . . . . . . . . . . . . . . .. 382

8.6. Conclusion........................................ 384 8.7. Appendix......................................... 384

8.7.1. :<\ppendix I-The Fundamental Isometry. . . . . . . . . . . . . . . .. 384 8.7.2. Appendix 2-Jitter Formulae. . . . . . . . . . . . . . . . . . . . . . .. 387 8.7.3. Appendix 3-An Exact Reconstruction. . . . . . . . . . . . . . . . .. 388 8.7.4. Appendix 4---Skip Sampling. . . . . . . . . . . . . . . . . . . . . . .. 388 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 390

Chapter 9. Zero-Crossings of Random Processes with Application to Estimation and Detection J. T. Barnett

9.1. Introduction....................................... 393 9.1.1. Expected Zero-Crossing Rate of a Gaussian Process. . . . . . . . .. 394

9.2. Discrete Frequency Estimation via Zero-Crossings. . . . . . . . . . . . . . .. 396 9.2.1. Higher Order Crossings. . . . . . . . . . . . . . . . . . . . . . . . . .. 398

9.3. The He-Kedem Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 398

Contents xix

9.3.1. Examples of Parametric Families and Contraction Mappings . . .. 402 9.4. Radar Detection via Level-Crossings of the Envelope Process . . . . . . .. 405

9.4.1. The Envelope of a Gaussian Process. . . . . . . . . . . . . . . . . .. 410 9.4.2. The Joint Density of R(t) and R'(t). . . . . . . . . . . . . . . . . . .. 411 9.4.3. The Squared Envelope Process. . . . . . . . . . . . . . . . . . . . .. 414

9.5. Level-Crossing Based Detector. . . . . . . . . . . . . . . . . . . . . . . . . .. 415 9.5.1. Variance of the Level-Crossing Count. . . . . . . . . . . . . . . . .. 415 9.5.2. Variance for the Envelope Process. . . . . . . . . . . . . . . . . . .. 417

9.6. Asymptotic Normality for the Level-Crossings of the Envelope of a Gaussian Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 426 9.6.1. Preliminaries................................. 428

9.7. Summary........................................ 434 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 436

Chapter 10. Magnetic Resonance Image Reconstruction from Nonuniformly Sampled k-Space Data F. T. A. W Wajer, G. H. 1. A. Stijnman, M. Bourgeois, D. Graveron-Demilly, D. van Ormondt

10.1. Introduction. .. 10.2. Sampling Strategies in MRI .

10.2.1. The Basic MRI Experiment. 10.2.2. Alternative k-Space Trajectories. 10.2.3. Encoding. 10.2.4. Sampling Summary

10.3. Image Reconstruction from Raw k-Space Data. 10.3.1. Introduction . 10.3.2. Radial Samples on a Square or Cube. 10.3.3. Gridding of Nonuniform Sampling Distributions . 10.3.4. Bayesian Image Reconstruction

10.4. Applications 10.4.1. Introduction . 10.4.2. Sparse Cartesian Sampling 10.4.3. Sparse Radial Sampling. 10.4.4. Spiral Sampling. 10.4.5. Random Sampling

10.5. Summary and Conclusions. Acknowledgments. References

439 440 440 442 444 444 445 445 445 448 454 460 460 461 464 466 469 473 474 475

xx Contents

Chapter 11. Irregular and Sparse Sampling in Exploration Seismology A. J. W. Duijndam, M. A. Schonewille and C. O. H. Hindriks

11.1. Introduction...................................... 479 11.1.1. Spatial Sampling in Exploration Seismology . . . . . . . . . . . .. 480 11.1.2. Subdomains................................. 480 11.1.3. General Strategy for Reconstruction . . . . . . . . . . . . . . . . .. 481

11.2. 1-D Reconstruction Using the Fourier Transform. . . . . . . . . . . . . . .. 482 11.2.1. Introduction................................. 482 11.2.2. Forward Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 484 11.2.3. MAP Estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 485 11.2.4. Weighting.................................. 485 11.2.5. Uniqueness and Stability. . . . . . . . . . . . . . . . . . . . . . . .. 487 11.2.6. Reconstruction Per Time Slice or Per Temporal Frequency

Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 488 11.2.7. Reconstruction in Space Versus Reconstruction in the Fourier

Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 489 11.2.8. Edge Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., 490 11.2.9. Signal Outside the Bandwidth. . . . . . . . . . . . . . . . . . . . .. 491 11.2.10. Uncertainty Analysis and Quality Control ............. , 491 11.2.11. Applications and Results . . . . . . . . . . . . . . . . . . . . . . .. 493 11.2.12. Field Data Example . . . . . . . . . . . . . . . . . . . . . . . . . .. 495

11.3. Two-Dimensional Reconstruction of Irregularly Sampled Signals. . . . . .. 496 11.3.1. Introduction................................. 496 11.3.2. Forward Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 496 11.3.3. Least Squares Inversion. . . . . . . . . . . . . . . . . . . . . . . . .. 497 11.3.4. Examples of 2-D Reconstruction. . . . . . . . . . . . . . . . . . .. 501

11.4. Sparse Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 502 11.4.1. Introduction................................. 502 11.4.2. Radon Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 505 11.4.3. Aliasing Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . .. 506 11.4.4. The Mixed Fourier-Radon Transform. . . . . . . . . . . . . . . . .. 508 11.4.5. Region of Support. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 509 11.4.6. Sampling in the q Domain. . . . . . . . . . . . . . . . . . . . . . .. 510 11.4.7. Results .................................... 511 11.4.8. Conclusions................................. 511

11.5. Efficiency of Reconstruction Algorithms. . . . . . . . . . . . . . . . . . . .. 512 11.5.1. Computational Complexity. . . . . . . . . . . . . . . . . . . . . . .. 514 11.5.2. Conjugate Gradient Schemes for General Irregular Geometries.. 515 11.5.3. Illustration of the CG Scheme for 1-D Fourier Reconstruction .. 517 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 517

Contents xxi

Chapter 12. Randomized Digital Optimal Control W. 1. De Koning and 1. G. van Willigenburg

12.1. Introduction...................................... 519 12.2. Optimal Control Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 521 12.3. Optimal Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 526 12.4. Optimal Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 533 12.5. Influence of Stochastic Sampling. . . . . . . . . . . . . . . . . . . . . . . .. 535 12.6. Conclusions...................................... 540

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 541

Chapter 13. Prediction of Band-Limited Signals from Past Samples and Applications to Speech Coding D. H. Mugler and Y. Wu

13.1. Introduction...................................... 543 13.2. Prediction from Past Samples Based on Nyquist-Type Criterion . . . . .. 545

13.2.1. Derivation of the Prediction Coefficients. . . . . . . . . . . . . .. 545 13.2.2. The Nyquist Condition and the Prediction Problem . . . . . . .. 555 13.2.3. The Discrete Prolate Spheroidal Sequences and Prediction. . .. 556

13.3. Prediction from Past Periodic Nonuniform Samples. . . . . . . . . . . . .. 557 13.3.1. Introduction................................ 557 13.3.2. DPSS for Interlaced Sampling. . . . . . . . . . . . . . . . . . . .. 561 13.3.3. Example of Prediction Using Interlaced Sampling. . . . . . . .. 570

13.4. Prediction from Past Samples Using Nonlinear or Other Methods . . . .. 572 13.5. Speech Signals - Prediction and Compression. . . . . . . . . . . . . . . .. 574

13.5.1. An Application of the Linear Prediction Methods to Speech Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 575

13.5.2. Comparison to Well-Known LPC Methods . . . . . . . . . . . .. 577 13.5.3. Linear Prediction and Data Compression. . . . . . . . . . . . . .. 578

13.6. Summary....................................... 582 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 583

Chapter 14. Frames, Irregular Sampling, and a Wavelet Auditory Model

J. J. Benedetto and S. Scott

Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 585 14.1. Introduction...................................... 585 14.2. Mathematical Background. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 587 14.3. Wavelet Auditory Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 589

14.3.1. Setting................................... 589

xxii Contents

14.3.2. Sigmoidal Operation. . . . . . . . . . . . . . . . . . . . . . . . . . .. 590 14.3.3. Lateral Inhibitory Network. . . . . . . . . . . . . . . . . . . . . . .. 592 14.3.4. The WAM Problem and Solution. . . . . . . . . . . . . . . . . . .. 592 14.3.5. WAM Wavelet Frame. . . . . . . . . . . . . . . . . . . . . . . . . .. 593

14.4. Theory of Frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 594 14.5. Irregular Sampling and Fourier Frames. . . . . . . . . . . . . . . . . . . . .. 601 14.6. Filter Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 606

14.6.1. Cochlear Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 606 14.6.2. Other Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 609

14.7. WAM Implementation and an Application to Speech Coding. . . . . . . .. 612 14.7.1. WAM Implementation. . . . . . . . . . . . . . . . . . . . . . . . . .. 612 14.7.2. An Application to Speech Coding . . . . . . . . . . . . . . . . . .. 614 Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 615 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 616

Chapter 15. Application of the Nonuniform Sampling to Motion Compensated Prediction for Video Compression A. Sharaf, F. Marvasti and M. Hasan

15.1. Introduction...................................... 619 15.2. A Review of the Previous Work on ST Based MC . . . . . . . . . . . . . .. 620

15.2.1. The Different Models of ST . . . . . . . . . . . . . . . . . . . . . .. 620 15.2.2. Backward versus Forward Mapping. . . . . . . . . . . . . . . . . .. 622 15.2.3. Estimation of the ST Motion Parameters . . . . . . . . . . . . . .. 624

15.3. Prediction Frame Reconstruction from Irregularly Spaced Samples . . . .. 627 15.3.1. The Nearest Voronoi Substitution Technique. . . . . . . . . . . .. 627 15.3.2. The Inverse-Distance Weighting Interpolation Technique. . . . .. 629 15.3.3. The Time Varying Technique . . . . . . . . . . . . . . . . . . . . .. 629 15.3.4. The Iterative Technique. . . . . . . . . . . . . . . . . . . . . . . . .. 630

15.4. A Fast BST MC Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 631 15.4.1. The MC Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . .. 631 15.4.2. The Fast-Search Refinement Algorithm of Estimated ST Motion

Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 634 15.5. Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 637

15.5.1. The Test Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . .. 637 15.5.2. The Statistics of the Irregular Sampling Process Resulting from

the Use of Forward Mapping BST for MC . . . . . . . . . . . . .. 638 15.5.3. Comparison of Prediction PSNRs. . . . . . . . . . . . . . . . . . .. 639 15.5.4. Comparison of Computational Complexities of the Various

Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 643 15.5.5. Conclusions................................. 645 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 646

Contents xxiii

Chapter 16. Applications of Nonuniform Sampling to Nonlinear Modulation, AID and D I A Techniques F. Marvasti and M. Sandler

16.1. Introduction...................................... 647 16.2. Pulse Position Modulation (PPM) ....................... " 648

16.2.1. Analysis and Recovery in L2 . . . . . . . . . . . . . . . . . . . . .. 649 16.2.2. Nyquist Rate in PPM. . . . . . . . . . . . . . . . . . . . . . . . .. 650 16.2.3. Recovery Techniques Based on Sampling Theory. . . . . . . .. 650 16.2.4. Analysis and Recovery in Finite Dimensional eN . . . . . . . .. 650

16.3. Pulse Width Modulation (PWM) . . .. . . . . . . . . . . . . . . . . . . . .. 657 16.4. FM Demodulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 657

16.4.1. FM Spectral Analysis and Lagrange Interpolation (LI) . . . . .. 658 16.4.2. Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . .. 659

16.5. Sine-Wave Crossings (SWC). . . . . . . . . . . . . . . . . . . . . . . . . . .. 661 16.5.1. Analys~s for L1 Band-L~i~ed Si~als . . . . . . . . . . . . . . .. 663 16.5.2. AnalYSIS for e Band-Lllruted Signals. . . . . . . . . . . . . . .. 664

16.6. Delta Modulation (DM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 668 16.7. Modulation Techniques for Data Conversion. . . . . . . . . . . . . . . . .. 671

16.7.1. Introduction................................ 671 16.7.2. Preliminaries................................ 672 16.7.3. Sigma Delta Modulation (SDM). . . . . . . . . . . . . . . . . . .. 674 16.7.4. Pulse Width Modulation (PWM). . . . . . . . . . . . . . . . . . .. 679 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 686

Chapter 17. Applications to Error Correction Codes F. Marvasti

17.1. Introduction..................................... 689 17.2. Erasure and Impulsive Channels. . . . . . . . . . . . . . . . . . . . . . . .. 692 17.3. Erasure Recovery Using Discrete Transform Techniques. . . . . . . . .. 692 17.4. Burst Error Recovery in an Erasure Channel Using Block Codes. . . .. 694

17.4.1. Introduction............................. . . 694 17.4.2. The Peterson Recovery Technique-RS Decoding . . . . . . .. 694 17.4.3. Forney's Decoding Algorithm. . . . . . . . . . . . . . . . . . . .. 697 17.4.4. Convolutional Methods for Erasure Channel. . . . . . . . . . .. 697

17.5. Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 698 17.5.1. Mathcad Simulations. . . . . . . . . . . . . . . . . . . . . . . . .. 698 17.5.2. DSP Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . .. 701 17.5.3. Objective Evaluation of the DSP RS Decoding

Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 702 17.5.4. Subjective Evaluation of the DSP RS Decoding

Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 703 17.6. Noise Sensitivity of Error Correction Codes in the Galois Field of

Real/Complex Numbers in an Erasure Channel . . . . . . . . . . . . . .. 704

xxiv Contents

17.6.1. Sensitivity to Quantization and Additive Noise. . . . . ..... , 705 17.6.2. Analysis.................................. 706 17.6.3. Forney's Method ............................ , 710

17.7. Other Transform Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . .. 711 17.8. Computational Complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 713

17.8.1. Lagrange Interpolation. . . . . . . . . . . . . . . . . . . . . . . . .. 714 17.8.2. The Conjugate Gradient (CG) Technique. . . . . . . . . . . . . .. 716 17.8.3. RS Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 716 17.8.4. Comparison of the Techniques. . . . . . . . . . . . . . . . . . . .. 716

17.9. Recovery in an Additive Impulsive Noise Environment. . . . . . . . . . .. 717 17.9.1. TheDFTCase .............................. 717 17.9.2. Application of Walsh Transform to Forward Error Correction.. 722 17.9.3. Discrete Cosine Transform (DCT). . . . . . . . . . . . . . . . . .. 726

17.10. Similarity of the Generator and Parity Check Matrices with Our Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 728

17.11. 2-D Erasure Recovery. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 729 17.11.1. Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . .. 731

17.12. A Comparison between Error Correction and Error Conceahnent. . . . .. 732 17.13. Conclusion...................................... 732

Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 734 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 735

Chapter 18. Application of Nonuniform Sampling to Error Concealment M. Hasan and F. Marvasti

18.1. Introduction...................................... 739 18.2. Novel Subimage Error Concealment Techniques. . . . . . . . . . . . . . .. 740

18.2.1. Subimage Decomposition. . . . . . . . . . . . . . . . . . . . . . .. 741 18.2.2. The Iterative Technique. . . . . . . . . . . . . . . . . . . . . . . .. 742 18.2.3. The Time Varying (Non-Linear Division) Technique. . . . . . .. 744 18.2.4. The Iteration With Overhead Technique. . . . . . . . . . . . . .. 744 18.2.5. The Iteration With DCT-Based Filtering Technique. . . . . . . .. 745 18.2.6. The Iterative/Time Varying Hybrid Technique. . . . . . . . . .. 746

18.3. The Recursive Error Concealment (REC) Technique. . . . . . . . . . . .. 749 18.4. The Matrix-Based Error Concealment (MEC) Technique. . . . . . . . . .. 752 18.5. Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 756

18.5.1. The Subimage Error Concealment Techniques. . . . . . . . . .. 760 18.5.2. The REC and MEC Techniques Applied to Isolated Random

Losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 766 18.5.3. The REC and MEC Techniques Applied to Isolated Block

Losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 773 18.5.4. The REC and MEC Techniques Applied to Consecutive Block

Losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 778

Contents xxv

Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 784 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 784

Chapter 19. Sparse Sampling in Array Processing S. Holm, A. Austeng, K. Iranpour and J.-F. Hopperstad

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 787 19.1. Introduction...................................... 788 19.2. Theory......................................... 789

19.2.1. Introduction to Array Processing. . . . . . . . . . . . . . . . . . .. 789 19.2.2. One-Way and Two-Way Beampattems . . . . . . . . . . . . . . .. 796 19.2.3. Random Arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 799 19.2.4. The Binned Random Array. . . . . . . . . . . . . . . . . . . . . .. 804

19.3. Optimization of Sparse Arrays . . . . . . . . . . . . . . . . . . . . . . . . .. 805 19.3.1. I-D Arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 809 19.3.2. I-D Layout Optimization. . . . . . . . . . . . . . . . . . . . . . .. 814

19.4. 2-D Array Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 818 19.4.1. Optimization of Steered Arrays . . . . . . . . . . . . . . . . . . .. 818 19.4.2. 2-D Array Optimized with Simulated Annealing. . . . . . . . .. 820 19.4.3. 2-D Array Optimized with Genetic Optimization. . . . . . . . .. 820

19.5. Optimization of the Two-Way Beampattem. . . . . . . . . . . . . . . . . .. 823 19.6. Conclusion...................................... 826

Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 828 Appendix: Optimization Methods. . . . . . . . . . .. . . . . . . . . . . . .. 829 A. Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 829 B. Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 830 C. Genetic Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 831 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 831

Chapter 20. Fractional Delay Filters-Design and Applications V Valim~i1ci and T. I. Laakso

Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 835 20.1. Introduction...................................... 836 20.2. Ideal Fractional Delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 837

20.2.1. Continuous-Time System for Arbitrary Delay. . . . . . . . . . .. 837 20.2.2. Discrete-Time System for Arbitrary Delay. . . . . . . . . . . . .. 838 20.2.3. Fractional Delay and Signal Reconstruction. . . . . . . . . . . .. 839 20.2.4. Characteristics of the Ideal Fractional Delay Element . . . . . .. 840

20.3. Design of FIR Fractional Delay Filters. . . . . . . . . . . . . . . . . . . . .. 842 20.3.1. Polyphase FIR Filters .............. : . . . . . . . . . . . 843 20.3.2. Least Squared Integral Error Design . . . . . . . . . . . . . . . .. 845

xxvi Contents

20.3.3. Weighted Least Squared Integral Error FIR Approximation of a Complex Frequency Response. . . . . . . . . . . . . . . . . . . .. 848

20.3.4. Maximally-Flat FIR FD Design: Lagrange Interpolation. . . . .. 849 20.3.5. Minimax Design of FIR FD Filters. . . . . . . . . . . . . . . . . .. 851 20.3.6. Fractional Delay Filters Based on Splines. . . . . . . . . . . . . .. 851 20.3.7. General Properties of FIR FD Filters. . . . . . . . . . . . . . . . .. 852

20.4. Design of IIR Fractional Delay Filters. . . . . . . . . . . . . . . . . . . . . .. 853 20.4.1. Discrete-Time All-Pass Filter . . . . . . . . . . . . . . . . . . . . .. 854 20.4.2. Design of All-Pass Fractional Delay Filters. . . . . . . . . . . . .. 855 20.4.3. Discussion.................................. 859

20.5. Time-Varying Fractional Delay Filters. . . . . . . . . . . . . . . . . . . . . .. 859 20.5.1. Consequences of Changing Filter Coefficients . . . . . . . . . . .. 860 20.5.2. Implementing Variable Delay Using FIR Filters . . . . . . . . . .. 861 20.5.3. Time-Varying Recursive FD Filters. . . . . . . . . . . . . . . . . .. 866 20.5.4. Conclusions................................. 868

20.6. Fractional Delay Filters for Nonuniformly Sampled Signals. . . . . . . . .. 870 20.7. Applications of Fractional Delay Filters. . . . . . . . . . . . . . . . . . . . .. 873

20.7.1. Sampling-Rate Conversion. . . . . . . . . . . . . . . . . . . . . . . . 873 20.7.2. Synchronization in Digital Receivers. . . . . . . . . . . . . . . . .. 875 20.7.3. Music Synthesis Using Digital Waveguides. . . . . . . . . . . . .. 876 20.7.4. Other Musical Applications of Fractional Delays. . . . . . . . . .. 878 20.7.5. Interpolated Array Beamforming . . . . . . . . . . . . . . . . . . .. 879 20.7.6. Design of Special Digital Filters . . . . . . . . . . . . . . . . . . .. 880

20.8. Conclusions...................................... 883 Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884 MATLAB Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 885 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 885

Author Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 897

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 901

CD-ROM Contents . ................................... 919